Some relationships with subnormal operators and existence of hyperinvariant subspaces

Key Points

• If a polynomially bounded operator has a specific invariant subspace, then it possesses a nontrivial hyperinvariant subspace.
• The operators involved must include both a unilateral shift and a subnormal operator in orthogonal complement for this outcome.
• Intertwining a polynomially bounded operator with hyponormal and its adjoint also ensures the existence of a nontrivial hyperinvariant subspace.
• The direct study of hyperinvariant subspaces specific to subnormal operators is not the focus here.